A numerical solution approach for non-smooth optimal control problems based on the Pontryagin maximum principle

TL;DR

This paper develops a numerical method for solving nonsmooth optimal control problems governed by elliptic PDEs, based on the Pontryagin maximum principle, and confirms its effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces a novel optimization approach leveraging the Pontryagin maximum principle for nonsmooth PDE-constrained control problems.

Findings

Discrepancy in the maximum principle diminishes along the iterative sequence.

Numerical experiments validate the convergence and effectiveness of the proposed method.

Abstract

We consider nonsmooth optimal control problems subject to a linear elliptic partial differential equation with homogeneous Dirichlet boundary conditions. It is well-known that local solutions satisfy the celebrated Pontryagin maximum principle. In this note, we will investigate an optimization method that is based on the maximum principle. We prove that the discrepancy in the maximum principle vanishes along the resulting sequence of iterates. Numerical experiments confirm the theoretical findings.